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Nonlinear Schrödinger equation and the Bogolyubov-Whitham method of averaging. (English. Russian original) Zbl 0639.35013

Theor. Math. Phys. 71, 584-588 (1987); translation from Teor. Mat. Fiz. 71, No. 3, 351-356 (1987).
Summary: An averaging of the type of B. A. Dubrovin and S. P. Novikov [Sov. Math., Dokl. 27, 665-669 (1983); translation from Dokl. Akad. Nauk SSSR 270, 781-785 (1983; Zbl 0553.35011)] is investigated for the nonlinear Schrödinger equation using the technique of finite-gap averaging [see S. P. Novikov, Usp. Mat. Nauk 40, No.4(244), 79-89 (1985; Zbl 0603.76001) and G. B. Whitham, “Linear and nonlinear waves” (1974; Zbl 0373.76001)]. For the single-gap case, the results are given explicity. Some characteristics of the original equation needed for applied calculations are averaged. Finally, recursion and functional formulas connecting the densities of the integrals of the motion of the Schrödinger equation, the fluxes, and the variational derivatives are given.

MSC:

35G20 Nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
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References:

[1] B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR,270, 781 (1983).
[2] S. P. Novikov, Usp. Mat. Nauk,40, 79 (1985).
[3] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York (1974).
[4] S. P. Novikov (ed.), The Theory of Solitons [in Russian], Nauka, Moscow (1980). · Zbl 0598.35003
[5] F. Flaschka, M. G. Forest, and D. W. McLaughlin, Commun. Pure and Appl. Math.,33, 739 (1980). · Zbl 0454.35080
[6] M. G. Forest and D. W. Mclaughlin, ?Modulations of sin-Gordon and sine-Gordon wavetrains,? Preprint (1982). · Zbl 0541.35071
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